Annals qfGlaciology 27 1998 © International G laciological Society

Glacial isostasy and the crustal structure of Antarctica

C. ZWECK

Antarctic GRG and IASOS, Universiry of Tasmania, Box 252-80, Hobart, Tasmania 7001, Australia

ABSTRACT. Tradi tional models of g lacia l isostasy, derived predominantly for study­ing the response of the ear th to the retreat of Northern H emisphere ice sheets since the last ice age, use earth models which assume constant lithospheric thickness. For Antarctica, where the continent below the ice sheet is two separate land masses differing in geological form, the assumption tha t a uniform li thosphere can expla in isos tatic behaviour is ques­tionable. H ere, a method to calculate the glacio-isostatic adjustment of the continent with a laterally varying li thospheric thickness is presented. The method is then used in a time­dependent ice-sheet model to model the isostatic response of the continent when the ice sheet passes through a glacia l/interglacia l transition. Various relati ons between the "crus­ta l thickness" beneath Anta rctica, derived from seismic data, a nd the "Ii thospheric thick­ness" estimate used in g lacio-isostatic calculations are assumed in a sensitivity study. Using the simples t relationships between crustal and lithospheric thickness, the greatest sensitiv­ity of the ice sheet to the crustal structure of Antarctica is not in the interior but at coastal locations, particul a rly near the maj or ice shelves. During periods of ice-sheet adva nce the grounding-line migrat ion of the shelves varies accordi ng to the depression of the earth peripheral to the ice sheet. T he peripheral depression depends on the regional elas ticity of the lithosphere which is controlled by the lithospheric thickness. Therefore the capacity for the ice sheet to advance vari es with the regional thickness of the crust.

1. INTRODUCTION

One of the primary motivations for studying the uplift of

formerly glaciated regions such as North America is to de­

duce a radial structure and viscosity profil e of the inner earth. T he disappearance of ice and subsequent isostatic response recorded as changes in rela tive sea level at coasta l locations is one ora limited set of phenomena that allow in­vestigation of the earth at depth . However, the process of

glacial isostasy is a lso important in the growth and decay of ice sheets themselves since changes in the shape of the earth modulate their behaviour. Le M eur and Huybrechts (1996) suggest that ice sheets a re affected by the isostat ic response of the earth through both the ice-sheet elevation (which affects surface accumulation) and elevation gradi­

ent (which affects the ice dynamics ). For marine-based ice sheets such as West Antarctica, Payne a nd others (1989) sug­gest that ice-shelf calving at the edge of the ice sheet is also affected by isostasy through the migrat ion of the g rounding line relative to sea-level height. Mass input, flu x and output a re all affected by the process ofisosta tic adjustment, and an

accurate model of glacial isostasy is crucial to the accurate simulation of ice-sheet behaviour.

Since the pioneering work of H askell (1935), geophysical studies of glacial isostasy have ass umed a lateral homogene­ity in the structure of the earth. Following the work ofPeltier (1974) and Cathles (1975), three latera lly uniform but

radially dependent earth-model parameters have been used as the basis of modelling attempts. These parameters a re the lower and upper mantle viscosities a nd the thickness or "rigidity" of the lithosphere. For the ice sheets of the Last G lacia l M aximum in the Northern H emisphere the later­ally homogeneous model of the earth explains the m aj ority of relative sea-level data. For Anta rctica, however, Stern a nd

Ten Brink (1989) sugges t that the ass umption of latera ll y uniform lithospheric thickness is inappropriate where East and 'Nest Antarctica a re of different geological origin. They

contend that a more accurate model would have a litho­spheric thickness of factor 5 times less in West Anta rctica than in East Antarctica. Le M eur and Huybrechts (1996) suggest that with a reduced lithospheric thickness the isostat ic adjustment in West Anta rctica is more loca l than predicted by models with uniform Iithosphere thickness.

The purpose of this paper is to investigate the extent to which non-uni form Ii thospheric rigidity a ffects the isostatic adjustment of Anta rctica and associated ice dynamics during a period of glacial/interglacia l transition. Generally, spherical ha rmonic methodology, which does not easily ac­

commodate the possibility of vari able lithospheric thick­

ness, is used in isostatic adjustment modelling. A technique is presented by which a non-uniform thickness model can be implemented and compari sons made with a uniform thick­ness model.

2. METHODS

The d ifferenti a l equation governing the equilibrium defl ec­tion of the surface of the earth with a uniform lithospheric thickness overlying a viscous mantle under the weight of an applied load is (Nadai, 1963):

(1)

where q is the pressure of the applied load, Pmgcp is the re­storing buoyancy of the earth's m antl e with a surface defl ec­tion cp a nd density Pm.) a nd D J" is the "fl exura l rigid ity"of the lithosphere. Equation (I) suggests that the weight of an ice sheet is compensated pa rti a ll y by the mantle and partiall y by the lithosphere. The degree of compensation depends on

321

Zweck: Glacial isostasy and crustal structure cif Antarctica

the spatial scale of the ice sheet and also on the value of the effective lithospheric rigidity which is a function of the lithospheric thickness:

(2)

where E is Young's modulus, H is the "effective elastic thick­ness" of the lithosphere and v is Poisson's ratio. Combining Equations (I) and (2), it can be deduced that traditional models of glacial isostasy predict that the magnitude of iso­static deflection 'P depends on the thickness of the litho­sphere, with a thin lithosphere allowing a greater deflection.

Equation (I) is the uniform-lithosphere case of the more general deflection equation:

(3)

When Dr is variable, Equation (3) is not a constant-coef­ficient partial differential equation and is therefore not sus­ceptible to Fourier solution. The most straightforward technique to solve Equation (3) is in coordinate space using sparse matrix methods. As an operator equation in finite difference form, Equation (3) becomes:

[\72 D i,j\72 + Pm9]'Pi,j = [A]'Pi ,j = qi,j . (4)

The isostatic deflection is the product of the inverse of matrix A and the applied load:

A- I 'Pi,j = qi ,j . (5)

From this equation the ultimate steady-state isostatic deflec­tion 'Pi.j for an arbitrary load qi,j can be determined. Time­dependent changes in the load q generate changes in the deflection 'Pi ,j which are used to evaluate the magnitude of isostatic disequilibrium and subsequent time-dependent isostatic response of the mantle. To generate A in the present work the nine-point finite difference form of \72 is taken from Abramowitz and Stegun (1965). Boundary conditions are imposed such that 'P and its first three spatial derivatives are set to zero at the model boundary. Although this arbi­trary choice of boundary conditions affects the hydro-iso­static component of adjustment, the effect is minimal away from the boundaries, while the present concern is with the ice sheet itself

A distribution of rigidity Di,j derived from lithospheric thickness by Equation (2) is required to generate A. Ander­son (1995) defines the "effective elastic thickness" H in Equa­tion (2) as the thickness of a uniform elastic plate that duplicates the flexural shape of the lithosphere on applica­tion of a geological load. In terms of this study, the defini­tion is problematic. The concept of the "effective elastic thickness" of the lithosphere has evolved from analyses of glacial isostasy that assume horizontal uniformity, so that the definition of effective elastic thickness is not guaranteed to be appropriate in the present study. However, in the study of Sabadini and Gasperini (1989) of non-uniform mantle viscosity it is shown that there is an order-of-magnitude si­milarity between isostatic adjustment predicted by models with uniform viscosity and that predicted by laterally het­erogeneous viscosity models. In this way it is not unreason­able to expect that the values for the uniform effective elastic thickness recovered from studies of glacial isostasy most likely represent a regionally averaged value. Breuer and Wolf (1995) and Kaufmann and Wolf (1996) estimated the magnitude of lateral variation of lithospheric thickness in the Svalbard archipelago near the northwestern corner of the Eurasian plate based on agreement with relative sea-

322

.~.

+

"

270' + .. 9()'

.. ~.

Fig. 1. Map cif crustaL thickness (km) for Antarctica, from Demenitskaya and Ushakov (1966).

level data. The extension of the definition of effective elastic thickness to that of a non-uniform plate which regionally duplicates the flexural shape of the load does not therefore seem unreasonable.

Figure I presents the crustal thickness of Antarctica as derived by Demenitskaya and Ushakov (1966) from gravity and seismic data. Although there is no clear relation between crustal and lithospheric thickness, the crustal values are used here to calculate the effective elastic thick­nesses of the lithosphere because they display an inherent lateral thickness structure and also reflect the high values of thickness in East Antarctica and low values in West Ant­arctica proposed by Stern and Ten Brink. It is possible that the variations in lithospheric rigidity reported by Stern and Ten Brink reflect variations in the elasticity of the crust and not its thickness. In this study, however, variations in litho­spheric rigidity are assumed to be caused by variations in lithospheric thickness. Two types of relation between crustal thickness and lithospheric thickness are assessed in the pres­ent work. First, the lithospheric thickness is assumed to be directly proportional to the crustal thickness. Second, the lithospheric thickness is assumed to be the crustal thickness plus a constant value. These are the simplest relations that can be used to generate realistic values for the effective elas­tic thickness while still retaining the qualitative differences in structure between East and West Antarctica suggested by Stern and Ten Brink. Neither of these relationships produces the quantitative differences in rigidity of Stern and Ten Brink. However, as the relationships used here between crust and lithosphere are somewhat tenuous, it is difficult to justify the use of more complex, power-law relationships.

To reflect glacial and interglacial conditions for the ice sheet, changes in both surface snow accumulation and eustatic sea level are imposed in a time-dependent manner. The imposed change in accumulation was derived using the technique of Budd and Smith (1982) using data from the Vostok ice core obtained by Jouzel and others (1987). The imposed eustatic sea-level changes were from Chappell and Shackleton (1986). This forcing drives a plan-view two-di­mensional dynamical ice-sheet model (details in Budd and

Q) (,)

29r----.----~--_.----_r----._--_,----~--_,

28.5 \

28

\

\ -UNIF ._- CT2 -CT3

CT4

~ 26 §

25.5

25

24.50L----2-'-O----~40----~6~O ----::8.LO----1-'-OO=----1~2c:-O --~~---:-: time (ka BP)

Fig. 2. Time-dependent change in total ice volume for CT

variable-lithosphere models. Uniform-rigidity model is also shown as the solid line. Major deviations in ice volume occur for CT4 between 120 and 70 ka BP (dotted line) and for CT2 between 55 and 0 ka BP ( dot -dashed line).

J enssen, 1989) with 100 km resolution coupled to an isostatic model with non-uniform lithospheric thickness described above. The time-dependent isostatic adjustment is modelled as a decoupled viscoelastic uniform-mantle-viscosity half­space as described in Cathles (1975). The mantle viscosity is 2 x 1021 Pa s, which is a factor of 2 greater then the custom­ary 1 x 1021 Pa s. McConnell (1968) notes that a model with decoupled viscoelastic rheology overestimates the elastic contribution and requires an increase in mantle viscosity to reproduce realistically the observed isostatic adjustment in the Northern Hemisphere.

3. RESULTS

Two relations between crustal thickness and lithospheric thickness are used in the present sensitivity study. The first is linear, and three particular cases are considered where the lithospheric thickness is 2, 3 and 4 times the crustal thickness illustrated in Figure I. The lithospheric rigidities of these cases span the customary 1025 N m value of uniform rigidity estimates, and are referred to here as CT2, CT3 and CT4. Table 1 shows the main statistical features of each of the models. When the lithospheric thicknesses have been gener­ated they are used in Equation (2) to compute correspond­ing lithospheric rigidities, and in Equation (4) to generate the matrix A.

The time-dependent changes in total ice-sheet volume generated by the ice-sheet model over a 160 ka glacial/inter­glacial simulation for CT2, CT3 and CT4 are shown in Fig­ure 2 alongside that of a uniform 1025 N m rigidity model referred to as UNIF. It is the time-dependent deviations in ice volume from UNIF that are of interest, and two major deviations are noted. CT4 has 1 x 106 km 3 more ice than a ny of the other models between 120 and 80 ka BP. After 80ka BP the difference disappears and the ice volume gener­ated by CT4 is of similar magnitude to that generated by the other models. The other major difference in ice volume, of I x 106 km 3

, occurs for the CT2 model from 55 ka BP to the present day.

Both of these ice-volume anomalies are positive and

<week: Glacial isostasy and crustal structure of Antarctica

~,

,.

270' ...

"

"

Fig. 3. Geographical distribution if difference in ice-sheet thickness ( m) between CT4 and UNIF at 80 ka BR

commence during a period of increased accumulation and ice volume over the continent. Figure 3 shows the geogra­phical di stribution of differences in ice-sheet thickness at 80 ka between CT4 and UNIF. The major differences occur in the interior of West Antarctica and the Ronne and Amery Ice Shelves, Although CT4 is over 100 m thicker in West Ant­arctica, the major ice-volume anomaly results from differ­ences near the ice shelves of over 500 m. Figure 4 shows the grounding line for CT4 and for UNIF at 80 ka BP. It shows that, in the region where the ice-volume differences are greatest, CT4 "grounds" further out onto the continental shelf than UNIF. The areas where the ice-volume differ­ences are greatest occur just behind the grounding line.

~,

...

~,

270' +

.. ...

~.

Fig. 4. Grounding line at 80 ka Jor UNIF (thick line ) and CT4 (thin line). Major differences in the position if the grounding line occur where the ice-volume differences are maxzmum.

90'

323

.:(weck: Glacial isostasy and crustal structure if Antarctica

Figure 5 shows the ice-sheet-volume changes as a func­tion of time for three cases of the second type of relation between crustal and lithospheric thickness. The cases are labelled CP30, CP50 and CP70 and correspond to situa­tions where lithospheric thickness is derived by simply add­ing 30, 50 and 70 km to the crustal thicknesses of Figure 1. The maximum, minimum and average field values for these cases are also shown in Table 1. For reference the time-de­pendent ice-sheet volume generated by UNIF is also shown in Figure 5. The major ice-volume anomaly occurs for CP30, which displays a pattern of behaviour similar to that of CT4 with an ice-volume difference of I x 106 km 3

, but continues throughout the glacial period to 20 ka BP. Figure 6 shows the regional ice-volume anomaly, and Figure 7 the ground­ing-line differences for CP30 and U IF at 80 ka BP. The ice­volume anomaly for the Amery Ice Shelf is not present for this model run, but there are significant differences near the Ronne Ice Shelf. Figure 7 shows that CP30 has grounded further out onto the continental shelf than UNIF. The pat­tern of ice-volume anomaly for the CP30 model is similar to that for CT2 and CT4.

4. DISCUSSION

In this study the greatest differences 111 ice volume from UNIF occur for the CT4, CT2 and CP30 models. Table I shows the average lithospheric rigidities of the lithosphere

24 for each model. Over a range of averages between 4.l x 10 and 4.1 x 1025 N m, large deviations in equilibrium deflec­tion profile rp and corresponding ice-sheet volume do not occur at large scale in the interior of East Antarctica. The general uniformity of behaviour in the central region be­

neath East Antarctica for the different lithosphere models can be understood by reference to the first-order equation for uniform lithospheric deformation in Equation (I). The standard two-dimensional Fourier transform is defined as (Sneddon, 1951):

j(kx, ky) = 1: 1: f(x, y)e- 27ri(kx X+kyy) dx dy (6)

where kx is the wavenumber in the x direction and ky is the

Q)

29.5 r----r----,-------,,-------,----,----r----,-----;

29

28.5

-UNIF - - CP30 -CP50

. CP70

.' \ I

E 27 ::>

~ Q) 26.5 .2

]! 26 B

25.5

25

24.50L---2.LO--4-"O,-----'60---:8':cO--' oLO'---'---'-:-20"---,:'l4-'-0---:-" 60

time (ka BP)

Fig. 5. Time-dependent change in total ice volumeJor GP vari­able-lithosphere models. Uniform-rigidity model is also shown as the solid line. Major deviation in ice volume occurs Jor GP30 between 120 and 20 ka BP.

324

+

Fig. 6. Geographical distribution if difference in ice-sheet thickness (m) between GP 30 and UNIF at 80 ka BP.

wavenumber in the y direction. Applied to Equation (I), the ratio of ice pressure to induced deflection is:

1 (7)

q D r k4 + PlUg ,----

where k = Jki + k~. The presence of D,. in the denomina-tor of EquatIOn (7) shows that the isostatic deflection cp is in­

versely dependent on the lithospheric rigidity. However, the importance of the spatial scale of the ice sheet is also appar­ent because of the presence of the wavenumber k in the de­nominator. Equation (7) suggests that for an ice sheet of

.~.

.Iet;.

270' ~ ... .. 90 "

+

Fig. 7. Grounding line at 80 ka for UNIF ( thick line) and GP 30 ( thin line). Major differences in the position if the grounding line occur where the ice-volume differences are maxzmum.

Table 1. Maximum, minimum and average values Jor the iffective elastic thickness ( H ) and lithospheric rigidity ( D r) qfthe Earth models used in the sensitivity study

M odel max ( H ) mill ( H ) max ( D r) mill ( Dr ) H Dr

km km Nm Nm km km

UNIF 88 88 [ X 1025 I X 1025 88 I X 1025

CT 2 102 45 2 X 1025 I X 1024 70 5.1 X 1024

CT 3 154 68 5 X 1025 4 X 1024 105 1.7 X [025

CT4 205 90 I X 1026 I X 1025 140 4.1 X 1025

CP30 82 53 8 X 1021 2 X 1024 65 4.1 X 1024

CP50 102 73 2 X 1025 6 X 102 1 85 9.1 x 1024

CP70 122 93 3 X 1025 I X 1025 105 1.7 X 1025

wavelength 850 km overlying a lithosphere of rigidity 1025 N m, D rk4 ~ Pmg, so that the ice is compensated to an equal degree by both the lithosphere and mantle. As the dominant mode of the wavelength spectrum of the Antarc­tic ice sheet is a few thousand kilometres, the effect of the lithospheric compensation term D rk

4 is very much less than Pmg, and the mantle compensation dominates. For the range of values spanned by the lithosphere models in this study, the value of effective lithosphere thickness is unimportant for large wavelengths. In the central region beneath East Antarctica the isosta tic defl ection is compensated domi­nantly by the mantle so that the lithospheric structure does not play a major role.

The spatial scale of the Antarctic ice sheet is so large that the magnitude of rigid ity does not affect the adjustment in the interior of East Antarctica. However, in '!\Test Antarctica and at the periphery of the ice sheet near the Ronne and Amery Ice Shelves the variable lithospheric model predicts an increased ice-sheet thickness and grounding-line extent compared to the model of uniform rigidity. The spatial scale of the anomalies between the uniform and non-uniform models is small compared to the entire ice sheet, in agree­ment with the low-pass-filter behaviour of the lithosphere discussed above. The last column in Table I shows that the models for which this increased ice volume occurs also have the la rgest deviation in average lithospheric rigidity from the uniform 1025 N m model. The mechanism by which the non-uniform lithospheric-thickness models ground out further onto the continental shelf than the uniform-thick­ness model can be understood as follows: During a period of increased accumulation the advancing ice sheet grounds by capturing the shallow sea fl oor to its front. The earth re­sponds to this grounding by a regional isostatic defl ection caused by the elastic lithosphere. The magnitude of the de­fl ection depends on the regional elasticity of the lithosphere. In particular, a lithosphere model with a large magnitude of regional rigidity produces a smaller defl ec tion then a model with small rigidi ty. As the defl ection at the ice front deter­mines whether further grounding can occur, the regional rigidity controls the extent of ice-sheet grounding.

Figure 8 illustrates thi s process in a simplified form using a radially symmetric parabolic-profil e ice sheet of Antarctic dimensions (3 km thickness and 1000 km radius) overlying a lithosphere of vari able but radially symmetrical thickness. The lithospheric-rigidity distribution is shown in the top panel, and the resulting equilibrium defl ection for each distribution is shown in the central panel. The isostatic

,(weck: Glacial isostasy and crustal structure qf Antarctica

I-J • / 0 1 0 -800L~ ________ ~~ ________ ~ ________ ~ __________ J

0.5 1 1.5

r~f ~:-,::-----/-~" :::~", - ,~- -- ' 1 _50L---------~.--------~--------~------~

0.5 1 1.5 Distance from ice centre ( x 103 km)

Fig. 8. Isostatic drflection Jor a parabolic-prqfile ice sheet with central height and spatial scale qf Antarctic dimensions over­lying litlzosphere qf varying thickness. (a) Rigidity distribu­tion Jor each lithosphere. ( b) Resulting diflection from the application qf a parabolic-prqfile ice sheet with 3 km central thickness and 1000 km radius. (c) Fractional diflection anomaly from the uniform-rigidity lithosphere model.

defl ection at the centre of the ice shee t is similar for all of the lithospheric structures, verifying that in the interior of the ice sheet the rigidity di stribution does not significantly affect the adjustment. However, a t the edge of the ice sheet larger deviations in the defl ection can be seen. The lower panel shows the deflection anomaly as a fractional differ­ence from the uniform-lithosphere case. For an ice sheet of this size the anomaly is over 50 m. The anoma ly for the lower-rigidity cases is positive at the edge of the ice sheet so that the bedrock elevation is higher there than for the uni­[orm-lithospheric-thickness model. This corresponds to a shallower sea fl oor then for the uniform-rigidity model, and an increased potential for further grounding by the ice sheet. The anoma ly is small , but, as the ice is already grounded nea r its edge, only slightly reduced deflections cause furth er grounding.

The analysis presented in Figure 8 ignores several fea­tures of the Antarctic ice sheet, such as the time-dependent isostatic adjustment, ice dynamics, hydro-isostasy and litho­spheric rigidities that do not have a constant spatial gradi­ent. However, as the lithosphere model is elastic, the deform ation pattern of deflection is smoothly varying so that the anoma ly from the uniform case is continuous. In this manner, in any region where the anomaly is positive the ice heet has a greater propensity for further grounding than in the uniform model.

CONCLUSIONS

Six different non-uniform lithospheric rigidity profiles have been used to assess the sensitivity of the Antarctic ice sheet to vari ations in lithospheric rigidity between East a nd West Anta rcti ca. The maj or differences in generated ice volume from the uniform-rigidity model occur in Wes t Antarctica and near the Amery a nd Ronne Ice Shelves, which are both major ice outflow shelves and shallower then the Ross Ice Shelf. The coasta l regions near these ice shelves are particu­larly sensitive to the variation in lithospheric rigidity. The

325

Zweck: Glacial isostasy and crustal structure rif Antarctica

main conclusion of this study is that the importance of non­uniform lithospheric rigidity in models of glacial isostasy and ice-sheet behaviour depends on the magnitude and variability of lithospheric thickness. The complex pattern of ice-volume changes is not easily understandable as a sim­ple function of the crustal relations used in this study. The most important question in this study is whether the rigidity distributions assumed in this study are realistic. Stern and Ten Brink's regional estimate for East and West Antarctica would suggest that the profiles of rigidity used here are rea­sonable. Kaufmann and Wolf's work to resolve the lateral structure of the earth around the Svalbard archipelago con­cludes that the ability to resolve the differences in litho­spheric structure from relative-sea-Ievel data is highly sensitive to the ice-sheet deglaciation history assumed for the region, with lithospheric thicknesses between 0 and 200 km found to satisfy the relative sea-level data. The pres­ent work concludes that the ice sheet is sensitive to varia­tions in the lithospheric structure, while Kaufmann and

Wolf conclude that the inference of lithospheric structure is sensitive to variations in the ice sheet. Therefore the process of deducing one from the other would not appear straight­forward . However, with the results presented here the regions more sensitive to lithospheric variation and corres­ponding ice behaviour for the Antarctic ice sheet can be outlined. The West Antarctic ice shelves are thought to be primarily responsible for contributing towards an increase of ice volume in Antarctica during the Last Glacial Maxi­mum (Huybrechts, 1990) and are considered also to be the region of the ice sheet most sensitive to increases in CO2

levels. The present study suggests that models of glacial iso­stasy using uniform lithospheric thickness underestimate the ice volume generated at coastal margins near the major ice shelves. As the value oflithospheric rigidity used in these models has been derived from studies analyzing relative­sea-level data from predominantly continental regions, the sensitivity of the ice shelves to models of non-uniform litho­spheric thickness needs to be further examined.

326

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